Countless molecular dynamics studies have relied on available ion and water force field parameters to model aqueous electrolyte solutions. The TIP4P/2005 model has proven itself to be among the best rigid water force fields, whereas many of the most successful ion parameters were optimized in combination with SPC/E, TIP3P, or TIP4P/Ew water. Many researchers have combined these ions with TIP4P/2005, hoping to leverage the strengths of both parameter sets. To assess if this widely used approach is justified and to provide a guide in selecting ion parameters, we investigated the transferability of various commonly used monovalent and multivalent ion parameters to the TIP4P/2005 water model. The transferability is evaluated in terms of ion hydration free energy, hydration radius, coordination number, and self-diffusion coefficient at infinite dilution. For selected ion parameters, we also investigated density, ion pairing, chemical potential, and mean ionic activity coefficients at finite concentrations. We found that not all ions are equally transferable to TIP4P/2005 without compromising their performance. In particular, ions optimized for TIP3P water were found to be poorly transferable to TIP4P/2005, whereas ions optimized for TIP4P/Ew water provided nearly perfect transferability. The latter ions also showed good overall agreement with experimental values. The one exception is that no combination of ion parameters and water model considered here was found to accurately reproduce experimental self-diffusion coefficients. Additionally, we found that cations optimized for SPC/E and TIP3P water displayed consistent underpredictions in the hydration free energy, whereas anions consistently overpredicted the hydration free energy.